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Algebra II

Simplifying Rational Expressions

Lesson

A rational expression is a fraction with polynomials on top and bottom. Simplifying works like simplifying numeric fractions: factor, then cancel common factors.

Procedure:

  1. Factor the numerator completely.
  2. Factor the denominator completely.
  3. Cancel any factor that appears in both.

You can only cancel factors (things multiplied), never terms (things added). E.g. x+3x+5\frac{x+3}{x+5}does NOT simplify by canceling the x’s — those are terms inside a sum.

Worked example 1

x2+5x+6x+3\frac{x^2 + 5x + 6}{x + 3}

Factor the numerator:

(x+2)(x+3)x+3\frac{(x+2)(x+3)}{x+3}

Cancel (x+3):

=x+2= x + 2

Worked example 2

x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}

Factor both:

(x+3)(x−3)(x+2)(x+3)\frac{(x+3)(x-3)}{(x+2)(x+3)}

Cancel (x+3):

=x−3x+2= \frac{x - 3}{x + 2}

How to type your answer

Write the simplified expression. If it’s a fraction, use / with parens around any sums. Use the form (x-3)/(x+2) with smaller-constant factor first if the result is a quotient. If everything cancels to a polynomial, just write the polynomial. Examples: x+2, (x-3)/(x+2), (x+1)/(x+5).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x2+3x+2x+1\frac{x^2 + 3x + 2}{x + 1}

Problem 2

x2+5x+6x+2\frac{x^2 + 5x + 6}{x + 2}

Problem 3

x2−4x+2\frac{x^2 - 4}{x + 2}

Problem 4

x2−9x−3\frac{x^2 - 9}{x - 3}

Practice

Standard problems matching the lesson.

Problem 5

x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}

Problem 6

x2−25x2+7x+10\frac{x^2 - 25}{x^2 + 7x + 10}

Problem 7

x2+4x+3x2−1\frac{x^2 + 4x + 3}{x^2 - 1}

Problem 8

x2+2x−15x2−9\frac{x^2 + 2x - 15}{x^2 - 9}

Problem 9

x2−4x+4x2−4\frac{x^2 - 4x + 4}{x^2 - 4}

Problem 10

x2+6x+9x2−9\frac{x^2 + 6x + 9}{x^2 - 9}

Problem 11

x2−7x+12x2−16\frac{x^2 - 7x + 12}{x^2 - 16}

Problem 12

x2+x−6x2+5x+6\frac{x^2 + x - 6}{x^2 + 5x + 6}

Problem 13

x2−8x+15x2−9\frac{x^2 - 8x + 15}{x^2 - 9}

Problem 14

2x2+5x+3x2−1\frac{2x^2 + 5x + 3}{x^2 - 1}

Problem 15

x2+2x+1x+1\frac{x^2 + 2x + 1}{x + 1}

Problem 16

x2−6x+9x−3\frac{x^2 - 6x + 9}{x - 3}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x2−11x+28x2−16\frac{x^2 - 11x + 28}{x^2 - 16}

Problem 18

2x2+3x−2x2−4\frac{2x^2 + 3x - 2}{x^2 - 4}

Problem 19

3x2−12x2+5x+6\frac{3x^2 - 12}{x^2 + 5x + 6}

Problem 20

x2−3x−10x2−25\frac{x^2 - 3x - 10}{x^2 - 25}

Problem 21

x2−16x2+8x+16\frac{x^2 - 16}{x^2 + 8x + 16}

Problem 22

4x2−92x2+5x+3\frac{4x^2 - 9}{2x^2 + 5x + 3}